Notes on the LVP and CVP in $p$-adic Fields

Chi Zhang; Mingqian Yao

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arXiv · 2512.24207

Notes on the LVP and CVP in $p$-adic Fields

Chi Zhang·Mingqian Yao

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Abstract

This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in $p$-adic fields. Leveraging the non-Archimedean property of $p$-adic norms, we propose a polynomial time algorithm to compute orthogonal bases for $p$-adic lattices when the $p$-adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and $p$-radicals in extension fields of $\mathbb{Q}_{p}$ to efficiently construct uniformizers and residue field bases, enabling rapid solutions for the LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over $\mathbb{Q}_p$.

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BibTeX RIS

Chi Zhang, Mingqian Yao. 2025-12-30. Notes on the LVP and CVP in $p$-adic Fields. https://arxiv.org/abs/2512.24207

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